Chapter 8.5, Problem 14E

a.

To determine

Explain to approximate the force against the end of the tank by a Reimann sum.

Answer to Problem 14E

  ${\displaystyle \sum _{i=1}^{k}62.4\cdot 2x_i\cdot \sqrt{3^2\cdot \left(1-\frac{x_i^2}{8^2}\right)}}dx$ approximately the force against the end of the tank by a Reimann sum.

Explanation of Solution

Given information: The vertical end of a tank containing water weighing $62.4lb/f{t}^3$ .

Shape- semiellipse.

  

Calculation:

The force of the water against the end of the tank can be approximated by diving the height into k partitions and evaluating the Riemann sum,

  ${\displaystyle \sum _{i=1}^{k}62.4\cdot 2x_i\cdot \sqrt{3^2\cdot \left(1-\frac{x_i^2}{8^2}\right)}}dx$

Hence, ${\displaystyle \sum _{i=1}^{k}62.4\cdot 2x_i\cdot \sqrt{3^2\cdot \left(1-\frac{x_i^2}{8^2}\right)}}dx$ approximately the force against the end of the tank by a Reimann sum.

b.

To determine

To find the force as an integral and evaluate it.

Answer to Problem 14E

The required solution is 7987.2.

Explanation of Solution

Given information: The vertical end of a tank containing water weighing $62.4lb/f{t}^3$ .

Shape- semiellipse.

  

Calculation:

As per the given problem,

Which by taking the limit as $k\to \infty$ , gives us the integral,

  $\begin{array}{l}{\displaystyle {\int }_0^{8}62.4\cdot 2\cdot x\sqrt{3^2\left(1-\frac{x^2}{8^2}\right)}dx}\\ =7987.2\end{array}$

Hence, the required solution is 7987.2.